Improvement in philosophical instruments or estimators



ATTUBNEYS.

M. STAPFF.

Patented Nov.24,1874.

Philosophical Instruments or Estimators."

UNITED STATES PATENT OEEICE.

FREDRIG MAURICE STAPFF, OF STOCKHOLM, SWEDEN.

IMPROVEMENT IN PHILOSGPHICAL INSTRUMENTS OR ESTIMATORS.

Specilication forming p art of Letters Patent N0. 157,239, datedNovember 24, 1874; application tiled October 4, 1873.

To all whom it may concern:

' Be it known that I, FEEDnrc MAURICE STAPFF, of the city of Stockholm,in the Kingdoln of Sweden, have invented a new and Improved Estimator,of which the following is a specification:

The estimator is a sliding rule, by which the volume oi prismatoidalbodies is calculated mechanically. As mo'st of the einbankments,ditches, cuts, 85o., occurring in the construe tion of railroads,canals, fortiiications, Ste., possess prismatoidal shape, the estimatorhas .the power to abridge and facilitate the important but tedious taskof computing` the quantities in earthwork. This task is facilitated bytables, which (in America) are based upon the prismatoidal formula, viz:

rf e E HIL: a {TID-at Making use of those two fundamental i'ormulas inthe construction of my instrument, I have found it proper to transformthe first one of them as follows:

If the volume sought has to be expressed by cubic yards, While the sizesof prisinatoid are measured by foot, this formula reads:

rI he following description, with drawing, refers to an estimatorconstructed in accordance with this last formula. But it should. beremembered that the arrangement of my instrument by no means needs to bechanged, and only three ot' its eight scales have to be modi- :Iied ifthe proportion between unity of length and unity of volume be anotherone than that of foot to cubic yard.

Vihen compared with the tables now in use, the estimator offers thefollowing advantages: It may be applied with equal ease, whatever(within the extent of the estimator-scales) the values of B and aentering the calculation may be, while the said tables are made up butfor certain oi'teuoccurring values oi' B and a it allows to nish thecalculation with more exactness and speed, and with less mental ei'-i'ort, than the tables do; it may be taken out to the field in agreat-coat pocket, while an equivalent complete collection of tableswould represent a little library; iinally, one estimator worth about $20will, to each railroadoffice, save the expense for one or more assistanis.

On the drawing adjoined to this description, and representing anestimator in reduced scale, the diiierent scales forming essential partsof the instrument are only generally indicated.

rlihe estimator is composed of they rule b, in which two slides, a andc, move, one above the other, parallel to the axis of the rule. Thelower slide a is provided with a diagram of curves, by means of whichthe average heights are found. The upper slide c carries iive parallelscales, (numbered 3 to 7,) and the rule b carries two scales (numbered land 2) at the one side, and one scale (numbered S) at the other side ofslidec. Of those scales, No. 7 and No. 8 are used for multiplicationsand arithmctical operations of similar nature. rllhe distances from thestarting-points of those scales to their did'erent graduated lines areproportional to the Briggs logarithms of the numerical values expressedby the respective graduated lines. I prefer ina-king the graduation ofone of those scales to progress from the left to the right, that of theother one in opposite direction.

For the special purpose of the estimator it is suilicient if themultiplication-scales represent the logarithms of numerical valuesbetween i).05 and 1000; but if the same scales should be used formultiplications, Cto., of

greater or smaller values, it is but necessary to suppose the graduatedlines to mean 1m or l, or ten times or one hundred times, the valuesprinted on the scales, and then to operate in the ordiiiaiy way.Besides, the extent of the graduation on those scales may be varied atpleasure.

For pcrforniin g iiiiiltiplicatioiis by means of scales 7th and Sth, theslide c should be drawn out until the graduated line indicating one ofthe factors on one of those scales is opposite that graduated line onthe other scale by which the other factor is indicated. rllie productthen is read on either of the two scales opposite the I line of theother one.

For working divisions the slide c should be drawn until the I line onone of the scales is in opposition with that graduated liiie on theother scale by which the dividend is indicated. Bead then thequotientfrom any of the scales opposite that graduated line on the otherscale by which the division is indicated.

Square roots are extracted by moving the slide c until the I line on oneof the scales is opposite to the graduated line on the other scaleindicating the power. The root is then indicated by those two graduatedlilies on both scales which coincide and express identical numericalvalues. A L-sliaped groove, 7.', ruiming along scale Sth in rule c,contains two buttons, B and n, which, in this groove, are movable to andfro. rlliese bnttoiis are made use of for indicating factors to bemultiplied by B or n.

The graduation of scale 2d (on rule D) and of scale 3d (on slide c) isidentical and uniform on both scales, the intervals between equivalentgraduated liiies being equal, but one of the scales (2d) is cipheredfrom the left to the right, the other one (3d) in opposite direction.

rlhe extent of the cipher series expressed on the addition-scalesdepends upon the extent of the work to be performed by the estimator. Ifthe sum ot' the heights of both terminal cross-sections (H+7z) does notexceed 100', it is sutlicient to express by each of the additioii-scalesthe cipher series 0 to l0() with subdivisions. c

For summing up two numerical values,f, i', H, and 7i, the slide c shouldbe dra-wn until the graduated line on scale 2d, expressing' one of thesums coincides with the graduated line by which on scale 3d the othersinn is expressed. rlhe sum then niay be read from either scale oppositethe Zero-line of the other one. The zero-line of scale 3d beingindicated by the edge of iiidex-plate d, (on slide 0,) is commonly usedto read the sums by.

If two values have to be summed the sinn of which is greater than 100and less than 200, the operation is quite the saine as here de. scribed;but to the value read from scale 3d, opposite the 100 line of scale 2d,100 must be added. For suniniin any two values with the assistance ofthe addition-scales, it is but necessary to understand the graduatedlines to mean l0 or 100 times tlie values printed on the scales, andthen to operate as described hereabove.

For effectingsubtractioiis by means of the estiniators addition-scales,the slide c should be drawn until the index-edge d meets with thatgraduated line on scale 2d by which the subtrahend is expressed. Beadthe rest from scale 3d or 2d, opposite that graduated line on 2d or 8dby which the subtractor is expressed.

By the scale 6th on slide c the function is expressed. The graduation ofthis scale is unii'orin, the intervals between equivalent degrees being5L times wider than those on scale 2d, where the sums H-l-i are producedas described above. The scale begins at the edge of iiidex-plate d,exactly below the Zero-line of scale 3d, proceeds from the right to thelelt, is ciphered in the same direction, and read by the edge ofindex-plate g, which is fixed on rule Z), exactly the zero-line of scale2d. Whatever the position of slide c (inside the rule Z1) may be, thedistance from the Zero-line of scale 2d to the index-plate d always mustbe equal to the distance between the index-plates d and g. Of course, ifany snm (H4-h) produced by lielp of the additionseales 2d and 3d is readfrom scale 2d by the edge of index d, synchronously the function H-l-L5i of index-plate g.

The subdivisions of scale 6th may be more or less extended, but, onestimators for coinn'ion use, I prefer to express on this scale directlyby graduated lines ,-166 c. yard.

By the scale 5th on slide c the function may be read from scale 6th bythe edge l is expressed. This scale proceeds from the edge ofindex-plate g; the distance from d to g, by which is indicated,

always being equal to the distance from the zero-point of scale 2d to d,by which (H-l-h) is indicated. You will read from scale 5th, by

T-S if you synchronously may iid IfI-i-L on scale 2d by index d.

The scale is constructed by the formula H-l-h: v/lllr, H-i-L meaning thedistance from d to the graduated line to be setout; a7, the value to beexpressed by that graduated line. rIlie subdivision ot' this scale maybe extended more or less; but, on estimators ot' common size, I preferto express directly, by graduated lines, l e. yards between 0 and 25; c.yards between 25 and the end of scale.

By the scale 4th the function is expressed. This scale begins at theedge of plate d, progresses from the right to thc left,

index g, the function and is read at the. edge of index-plate i, whichmay be drawn to and fro in the groove 7c, between the scales 1st and 2d.By means of the scale 1st (on rule b) this index can be placed so thatthe distance from it to edge of indexplate Zexpresses the differenceH-K-t, while the sum H+ h is synchronously expressed by the distancefrom g to d. Scale lst starts exactly above the zero-point of scale 2d,thence pro gressing to the right. Its graduation is a uniform one7 theintervals between equivalent graduated lilies being twice as wide as onscale 2d. Of course, if by index z' any value of 7a is indicated onscale lst, the value 2h will be synchronously indicated by the sameindex on scale 2d. If slide c is in such a position that the edge ofplate d coincides with any graduated line on scale 2d expressing thevalue H-l-k, and at the same time the index t' is in such a positionthat by it may be read h from scale lst, or 2h from scale 2d, then thevalue H+h-2r=Hh will be indicated by H-h 2 M bythe saine index on scale4th.

Scale 4th is constructed by the formula II-L: VMzIS 1,/ meaning thenumerindex on scale 3d, and the function ical value to be expressed byany graduated H I The value is read from scale 6th by index g, andmarked on scale 8th by the button B.

H 2 The value is read from scale 5th as read from scale 4th by index z'.The snm of both is marked on scale 8th by the button u.

The multiplication B (H 7L) is done by drawing slide c until thegraduated line on scale 7th which expresses the value B coincides withthe indicating-edge of button B. The product read from either of thescales 7th or 3th, opposite the I line of the other scale, is marked onscale 2d byindex t'. The multipli# by index g, and added to the valueVII-l-h 2 Eli-.7L 2 catlonux T8 w: :I is done by drawing slide c untilthe graduated line (marked by index z' on scale 2d) by last-foundproduct, a

The sum read from scale 2d by edge of index d v expresses, by cubicyards, the volume of one running foot of the prismatoid. Let the lengthL be 100, as commonly in America, where the distance between twostation-points is 100', the volume of the whole prismatoid is foundsimply by placing the decimal-mark of the sum next after the secondfractional cipher to the right hand; but, if the prismatoid extendingbetween intermediate or plus points be shorter than 100', themultiplication L x sum is easiest done with the assistance of themultiplication-scales 7th and Sth.

Lethbe 25'; H :.55; n 1%;13 22.5; L 100. After having marked 25 by indext on scale 1st, draw slide c until the graduated line 25 on scale 2dcoincides with the graduated line 55 on scale 3d. Bead from scale 6th,by the edge of index g, 1.48, and mark this value by index B on scale8th. Read from scale 4th, by the index z', 2.78, and from scale 5th, bythe edge of index g, 59.26. Add both values together and mark the sum,62.04, by index n on scale 8th. Draw slide c until the graduated line22.5 on scale 7th coincides with the indicating-edge of button B. Readthe product 33.3 opposite the I line of one scale Afrom the other one.and mark it by index t' on scale 2d. Draw slide c until the graduatedline 1.5 on scale 8th coincides with the indicatingedge of button a.Read the product 93.06 from either scale 7th or 8th opposite the I lineof the other one. Add both products by drawing slide c until thegraduated line 93.06 on scale 3d coincides with the graduated line33.30, as indicated on scale 2d by index z'. The snm 126.36, read fromscale 3d opposite the 100 line of scale 2d, is the volume (in cubicyards) of one ruiming foot of the prismatoid. The volume of the wholeprismatoid, being 100 in length, would be 126.36 cubic yards.

rIhe whole series of operations here described may, with some practice,easily be done in some few minutes, and in shorter time yet, if manycalculations have to be carried out in which the same values for n and Benter; or if the values to be operated upon are not too complicated 5 orif the crosssections are of triangular shape, (ditches j' c',) in whichcase, B

H Jf- L a l is dispensed with; or if the slope a is l, (common cuts,) inwhich case the multiplicabeing 07 the multiplication BX tion n isspared, or if il disappears.

Ff B z B' sion of the equation Hh n n n Ff for a given numerical valueof 7,-, and for B successive values of 7, the average height,

Hh, is expressed by the absciss distance to any point of curve theordinate of which is For constructing the curves, some numer- B icalvalues oi 27-, are set out as ordinates by arbitrary scale; and asabscisses are set out by the scale 2d or 3d, those values of Hh whichresult by substitution in the equa-tion above Ff of those values, 7 forwhich the curve has to be drawn, the several points of curve set out inthis way are connected by a continual line.

F The values for which succeeding curves are constructed, increase bysimple arithinetical progression. For the estimator here de- F' scribedthe increase of 7g is irom curve to Ff curve: one unit for valuesbetween 0 and l0; tive units between 10 and 100; ten units between 100and 1000; twenty units from 1000,

tween two succeeding values of may be,

, F intervals between two values expressed by succeeding curves arewider on the drawing than stated in the description.

For common practice it is sufficient if the curves are constructed forvalues 27, O 20, and for values Hh 0 50; but for the rest thecurve-diagram may be extended or abridged at pleasure, the exactiiess otthe estimator and the easiiiess of its -practice greatly increasing ifthe ordinates are set out by as large a scale as possible. I have takenthis scale four times greater than the width of slide would have allowedit' the undivided diagram had been iixed on it; and then I have cut thediagram in four ribbons parallelly to the axis of abscisses, whichribbons are iixed on both sides of slide a, the iirst ribbon containingthat portion of the-ciirves which com prises the ordinates 0 5; the sec-B B ondZ-n: 5. ..105 thethiri 27,: 10. ..105

the fourth;L l5 20 5 but I wish my patent to cover also estimators withundivided diagrams of curves.

The ciphered graduated lines or marks of degrees alongthe axis ofordinates, as represented on the drawing, express numerical values of721%; but lately I have constructed estimators where tlieciphers alongthe axis of ordinates indicate numerical values of while the respectivegraduated lines are constructed for values B il. Hereby the division aisavoideih bg.

ing operated upon directly.

Vhen making' use ofthe diagram oi' curves the slide c is drawn to thelett until the edge of index d coincides with the zero-line of scale 2d,then the slide ci is put in such a position as that the portion ofdiagram ot' curves which L contains the numerical value of E enteringinto calculation stands close before the plate d on slide c.

Along the plate d moves the tongue f, guided by a slot, at any point ofwhich it may be iixed by the brake-screw c. The curves are almostnormally transversed by the tongues oblique indicating edge, which isprovided with three index-lines. The slide a is in its right position ifthe most convenient one ot those index-lines meets the axis ot'ordinates of the diagram of curves. Then the tongue f should be moved upor down, and fixed so that the chosen index-line points out thegraduated line on the axis oi' ordinates by which the nu- L mericalvalue 7enterin g the calcula-tion is represented. If, now, slide a iskept in position, while slide c isv drawn to the right until the chosenindex-line on toiigne f meets that F curve by which the value of jentering the calculation is expressed, then the value ot' Hh, dependingon the respective values of F and g, may be read from scale 2d by indexd.

, F It any value of Tj has to be operated upon tonguef. Letfi jf beshould be drawn until the index-line on ton gue 175, the slide a curve170.

f points just in the middle between the curves 170 and 180. The spacebetween those curves,

until the index-line on tongue f points 1gdegree before curve 180, or 1%degree behind .E may The equationHi z [if i. 2

B F be written (H102 (Hh) %-f,or, generally,

x2 -l-Y a a0 b.

Of course, the estimator may be used directly for the solution of allsecond-degree equations of last-named shape, as far as the numericalvalues a (to be sought along the axis of ordinates) and b (expressed bythe curves) are inside the limits of the ciirve-diagram on slide a; ft a0. 40, and b 0 2500 4500, if the estimator is .of ordinary size.

F f B The divisions ,i haviiig been performed with or without the helpof the multiplication (division) scales 7th and 3th, that part of thecurve-diagram is placed before the index-plate d on slide c whichcontains the ordinate value B ,-L entering calculation. Previously, theslide c has been drawn to the left until the edge of index-plate l fallsin with the Zero-point of scale 2d, -and with the edge of plate g. The

tongue f is moved up or down until the index-line on same points outthat graduated line on axis of ordinates by which numerical B value ofenteringcalculation is expressed.

While slide a is kept in this position, slide c is drawn riglitwarduntil the index-line on tongue f meets that curve, or that point betweentwo ciirves, which represents the value The value of hbelonging to therespect- B ive values,-Z and may then be read from F point between twocurves, which represents Now, the value H-l-h could be read from scale2d by index d. But this reading can be (EL-702 spared, because you willnd, directly,

on scale 4th byindex i; T08- on scale 5th H by index g; on scale 6th byindex g.

These values, and those of u, B, L, are operated upon exactly asdescribed above for tlie case that H and li were given directly insteadof F and f. Given: F=4500Uf5 f=1500lIl 5 :3:22.55 71:15.

From the given values follows, directly,

Draw slide c until the zero-line of scale 2d is touched by theindicating-edge of d. Place right before slide c that part of slide awhich B contains the diagram for ordinate values 7,

f meets the curve 3f: 1000; then h 25 may be read from scale 2d by indexd, and kept in mind. Slide c being held in place, slide a is drawnrightward until the axis of ordinates is touched by the index-line ontongue f,- then slide a is held in place, while slide c is drawnrightward until the index-liiie on tongue f F touches the curve*7L 3000.From scale 2d could now (by index d) be read 70 78', this being H h.AThe value li 95', as previously read from scale 2d, now having beenmarked on scale 1st by index t, the following readings are done: Fromscale 6th, by index g, 1.35 is marked on scale 8th by index B; fromscale 4th, by index t', 1.60, and from scale 5th, by index g, 49.05. Thesum of both, or 50.65, is marked on scale 8th by button a'.

The multiplication B 1.35 is done by drawing slide c until the 22.5 lineof scale 7th is opposite the indicating-edge of button B. The product30.37, as read from either scale 7th or 8th, opposite the I line of theother one, is marked on scale 2d byindex i'. The multiplication a 50.65is done by drawing slide c until the 1.5 line of scale 7th is inopposition with the indicating-edge of button a. The product 75.97 isread from either scale 7th or Sth, opposite the I line of the other one.This product is added to the first one by drawing slide c until the75.97 degree of scale 3d coincides with the indicating-edge of index z'.The sum read from scale 3d opposite the 100 line of scale 2d, viz.,100+6.34=106.34, expresses the volume of each running foot of theprismatoid. i

If the second-degree equation x2+36-lw= 4205 has to be solved, the slidec should be drawn until the edge of index d coincides with the zero-lineof scale 2d., Then that piece of curve-diagram showing the ordinates 30i0 is placed before slide c, so that the index-line on tongue f meetsthe axis of ordinates, and the tongue is ixed so that its index-linepoints exactly between the ordinate degrees 36 and 30.2. Now the slide cis drawn (c meanwhile being held in place) until the index-line ontongue f is one-fourth ofthe distance between the curves 4200 and 4220behind the curve 4200. From scale 2d may now be read, by index fi, fc40.3.

Invertedly, the estimator may be used for dedncing mechanically from agiven volume the average height of the prismatoid containing thisvolume. Hereby the estimator proves very useful for determining how muchthe grade of a preliminary railroad-line ought to be attached, or howmuch such a line ought to be thrown to the side for balancing as much aspossible the quantities in the cuts and einbanliments ot'a givenrailroad-section, provided the ground on the sides ot' the preliminaryline previously has been crosssectioned. rlhe different scales may beturned so as to progress in a direction opposite the present one 5 orthe scale 1st may be applied on slide c, and synchronously the scales4th, 5th, 0th on rule b. Then the index-plate g is let't out, the scales4th, 5th, 0th being read by index d, and the index li is made to move ina groove on slide c; or the scale 1st maybe lei't out, the respectivereadings then to be done from scale 2d, which, for that purpose, shouldbe furnished with a second ciphering, identical with that on scale lst.Or the scales 4th and 5th may be combined, the spaces between equivalentgraduated lines being thrice as wide on scale 4th as they are on scale5th. 0r the whole instrument may be constructed in the shape of a prismor a cylinder, with the scales parallel to its axis, or in the shape ofa disk divided into degrees on its plan or edge, &c.

The drawing represents, in Figure 1, a plan view, and in Fig. 2 a headview, otthe estimator, Fig 3, a longitudinal, and Fig. a a transversal,section oi' same 5 Fig. 5, aview of slide a from both sides.

Z1, rule, in which, by double grooves, move the upper slide c and thelower side a, one above the other, d, index-plate attached to upperslide c; f, tongue sliding along slit ot' index-plate d.; c, brake-screwto iix that tongue with 5 1'., index sliding in groove L along one E,lA, index-buttons slidb 4 5 ingin groove k along other edge of slide c;g, index-plate attached to rule c; 7i., button for drawing slide c. (Maybe placed elsewhere on the slide c.)

The scales lst to Sth on rule b and slide c are on drawing, butindicated as mentioned in the ingress; likewise the diagram of curves onslide u.

Sums between 50 and 100 are found on this estimator by adding 50 to thevalue, as read from scale 3d, opposite the 50 line of scale 2d. Themultiplication-scales rth and 8th have such an extent as that factorsfrom 0.07 to l0 may by them be operated upon directly, and the diagramot' curves on the lower slide edge of slide c;

B 1 c 1s constructed for values w 0 40, Il:

625 0 GO, and corresponding ones ot H L=0 25. The graduatedordivisionlines along' the axis of ordinates of the diagram of curves onlower slide of estimator are ciphered by values of "Ii, as referred toin the description, while the corresponding degrees on drawings areciphered by values of l.

FREDRIC MAURICE STAPFF.

lVitnesses:

NEU. A. EFWING, EUGNE Fonswav.

